Non-commutative localisation and finite domination over strongly\n Z-graded rings

Let R be a strongly Z-graded ring with degree-0 subring S, and let C be a\nchain complex of modules over the subring P of elements of non-negative degree.\nWe show that there are non-commutative localisations of P which detect whether\nthe complex C is S-finitely dominated or S-contractible, respectively, and that\nthese localisations are universal among P-rings making S-finitely dominated and\nS-contractible complexes contractible. This generalises known results for\npolynomial rings to a much wider class of rings. We show by example that in\ngeneral C need not be P-homotopy finite even if C is S-finitely dominated; this\ndiffers from the case of polynomial rings.\n